Maximum principle for optimal control of McKean-Vlasov FBSDEs with Levy process via the differentiability with respect to probability law

Abstract

In this paper we study stochastic optimal control problem for general McKean-Vlasov-type forward-backward differential equations driven by Teugels martingales associated with some Levy process having moments of all orders and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions under which the necessary optimality conditions turn out to be sufficient. The proof of our main result is based on the differentiability with respect to probability law and a corresponding Ito formula.